Polarization of Ideal Gas
المؤلف:
Sidney B. Cahn, Gerald D. Mahan And Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 2 , p 20
3-9-2016
1669
Polarization of Ideal Gas
Calculate the electric polarization P of an ideal gas, consisting of molecules having a constant electric dipole moment p in a homogeneous external electric field E at temperature τ. What is the dielectric constant of this gas at small fields?
SOLUTION
The potential energy of a dipole in an electric field E is
where the angle θ is between the direction of the electric field (which we choose to be along the ẑ axis) and the direction of a the dipole moment. The center of the spherical coordinate system is placed at the center of the dipole. The probability dw that the direction of the dipole is within a solid angle dw = sin θ dφ dθ is
(1)
The total electric dipole moment per unit volume of the gas is
(2)
Introducing a new variable x ≡ cos θ and denoting α ≡ pE/τ, we obtain
(3)
where 
is the Langevin function. For pE << τ (α << 1), we can expand (3) to obtain
(4)
Since D = εE and
we have for the dielectric constant
(5)
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